Editing Bounded variation (section)

{{Short description|Real function with finite total variation}}
{{Use dmy dates|date=April 2023}}
In [[mathematical analysis]], a function of '''bounded variation''', also known as '''''{{math|BV}}'' function''', is a [[real number|real]]-valued [[function (mathematics)|function]] whose [[total variation]] is bounded (finite): the [[graph of a function]] having this property is well behaved in a precise sense. For a [[continuous function]] of a single [[Variable (mathematics)|variable]], being of bounded variation means that the [[distance]] along the [[Direction (geometry, geography)|direction]] of the [[y-axis|{{math|''y''}}-axis]], neglecting the contribution of motion along [[x-axis|{{math|''x''}}-axis]], traveled by a [[point (mathematics)|point]] moving along the graph has a finite value. For a continuous function of several variables, the meaning of the definition is the same, except for the fact that the continuous path to be considered cannot be the whole graph of the given function (which is a [[Glossary of differential geometry and topology#H|hypersurface]] in this case), but can be every [[Intersection (set theory)|intersection]] of the graph itself with a [[hyperplane]] (in the case of functions of two variables, a [[Plane (mathematics)|plane]]) parallel to a fixed {{math|''x''}}-axis and to the {{math|''y''}}-axis.

Functions of bounded variation are precisely those with respect to which one may find [[Riemann–Stieltjes integral]]s of all continuous functions.

Another characterization states that the functions of bounded variation on a compact interval are exactly those {{math|''f''}} which can be written as a difference {{math|''g''&nbsp;−&nbsp;''h''}}, where both {{math|''g''}} and {{math|''h''}} are bounded [[monotonic function|monotone]]. In particular, a BV function may have discontinuities, but at most countably many.

In the case of several variables, a function {{math|''f''}} defined on an [[open subset]] {{math|Ω}} of <math>\mathbb{R}^n</math> is said to have bounded variation if its [[distribution (mathematics)|distributional derivative]] is a [[Vector-valued function|vector-valued]] finite [[Radon measure]].

One of the most important aspects of functions of bounded variation is that they form an [[Associative algebra|algebra]] of [[continuous function|discontinuous functions]] whose first derivative exists [[almost everywhere]]: due to this fact, they can and frequently are used to define [[generalized solution]]s of nonlinear problems involving [[functional (mathematics)|functional]]s, [[ordinary differential equation|ordinary]] and [[partial differential equation]]s in [[mathematics]], [[physics]] and [[engineering]].

We have the following chains of inclusions for continuous functions over a closed, bounded interval of the real line:

: '''[[Continuously differentiable]]''' ⊆ '''[[Lipschitz continuous]]''' ⊆ '''[[absolutely continuous]]''' ⊆ '''continuous and bounded variation'''  ⊆ '''[[Differentiable function|differentiable]] [[almost everywhere]]'''